Law of small numbers

The law of small numbers, the two-thirds law or the law of the third is a set of stochastics, which describes a special case of the Poisson distribution. This term is usually used in connection with the game of roulette and describes the fact that, where 37 games about two -thirds of the 37 numbers are taken.

The term law of small numbers goes back to the Russian-German mathematician Ladislaus of Bortkewitsch (1898 ), who found that law in the study of the numbers of deaths caused by hoof beats in the various cavalry units of the Prussian army.

The law of small numbers in roulette

Looking at roulette several rotations, ie series of each of 37 individual matches ( coup ), it is found that over the course of a rotation only about two -thirds of the numbers are taken, about half of them several times, while the remaining third is not taken - hence the terms used by roulette players names two-thirds law or the law of the rare third.

During a rotation at roulette are on average

• Not taken 36.3 % of the numbers, that is, 13.4 numbers
• , Ie 13.8 numbers exactly once met 37.3 % of the numbers
• Made 18.6% of the numbers, that is, 6.9 numbers exactly twice
• Made 6.0% of the numbers, that is, 2.2 numbers exactly three times
• 1.7% of the numbers, ie numbers 0,6 or four times more often met.

NOTE: These are the exact values ​​, which were calculated using the binomial distribution; due to rounding, the totals do not result in exactly 100 % and 37 numbers.

According to the law of large numbers occurs in the long-term means each of the 37 numbers with the same relative frequency, ie, the number of coups is sufficiently large, as is attributable to each individual number of the same number, ie 1/37 = 2.7%. Looking at several rotations and a certain number in advance, so this is taken on average, in each rotation once.

This led many players to the fallacy that in a series of 37 coups each number occurring once. But this is not the case; rather, it is extremely unlikely that any number is hit exactly once; the probability of this happening is only 1.3 × 10-15.

In spite of the equal probability of all the numbers occurs in the case of a small number of games not a uniform distribution, but the probability distribution of the above predetermined pattern.

Also with the help of the two-thirds law can be no winning strategy, see (see March ).

The general case

The law of small numbers is a simple application of the Poisson distribution for λ = 1 and not only applies to rotations at roulette, but for any series of n independent games, which can take each n equally probable outputs. For example, if n objects will be raffled among n recipients and the individual draws are independent.

The law of small numbers is all the more accurate the larger the number n. For aiming the proportion of recipients to receive exactly k objects against the value

The proportion of recipients who get nothing, thus tends to 1 / e ≈ 36.7879 %. The same is true for the proportion of those who are considered exactly once.

The specifications given in the previous section numbers for n = 37 differ by only 0.5 % of the calculated using the Poisson distribution limits.

As rice grains

The picture on the right shows randomly scattered on the ground lying rice grains. Picture and grid size are chosen so that on average a square falls a grain of rice, that is, it is λ = 1

The counting of frequencies confirmed despite the small sample size of n = 64 (using the Poisson distribution approximate ) expectation values ​​:

• 23 squares contain no grain of rice. Expected value ( to 2 decimals rounded): 23.54
• 25 squares containing exactly one grain of rice. Expected value: 23.54.
• 12 squares contain exactly two grains of rice. Expected value: 11.77.
• 2 squares contain exactly three grains of rice. Expected value: 3.92.
• 2 squares containing four or more rice grains ( 1 x 4 or 1 x 5). Expected value: 1.22.

( The sum of the expectation values ​​is rounded to one decimal place: 64.0. )